Inequality (Quadratic and Linear)

An inequality is a mathematical expression showing the relationship between two expressions that are not necessarily equal. Instead of “=”, we use symbols like:

• “<” (less than)

• “>” (greater than)

• “≤” (less than or equal to)

• “≥” (greater than or equal to)

These are used to compare values and solve problems involving ranges of possible solutions.

🔷 Types of Inequalities in Quantitative Aptitude

1. 👉 Linear Inequalities (First Degree)

2. 👉 Quadratic Inequalities (Second Degree)

3. 👉 Compound and System Inequalities

4. 👉 Word-based Inequality Statements (common in Reasoning)

Let’s break down each.

🔶 1. LINEAR INEQUALITIES

Definition: An inequality involving variables raised only to the power 1.

🧮 Format: ax + b < c or ax + b ≥ c

✅ Solving Method:
Solve like a linear equation. Remember:

• When multiplying or dividing both sides by a negative number, flip the inequality sign.

🧩 Example 1:
Solve: 2x + 5 < 15
⟹ 2x < 10
⟹ x < 5
✅ Answer: All values of x less than 5 satisfy the inequality.

🧩 Example 2:
Solve: -3x + 4 ≥ 1
⟹ -3x ≥ -3
⟹ x ≤ 1 (Note: sign flipped when dividing by –3)

🔶 2. QUADRATIC INEQUALITIES

Definition: Inequalities where the highest degree of the variable is 2.

🧮 Format: ax² + bx + c > 0, or ≤ 0, etc.

✅ Steps to Solve:

1. Bring all terms to one side: ax² + bx + c (inequality sign) 0

2. Find the roots of the equation ax² + bx + c = 0

3. Draw a number line and mark the roots

4. Test intervals between roots to determine sign

5. Write the solution set based on the inequality

🧩 Example 3:
Solve: x² – 5x + 6 < 0

Step 1: x² – 5x + 6 = 0
⟹ Roots: x = 2 and x = 3
Step 2: Plot on number line:
Split into intervals: (–∞, 2), (2, 3), (3, ∞)

Test sign in each interval:

• Between 2 and 3 (say x = 2.5):
  x² – 5x + 6 = 6.25 – 12.5 + 6 = –0.25 < 0 → true
  ✅ Answer: 2 < x < 3

🧩 Example 4:
Solve: x² + 4x + 3 ≥ 0

Step 1: Factor → (x + 1)(x + 3) ≥ 0
Roots: x = –1, –3
Interval testing:

• (–∞, –3) → say x = –4 → (+)(+) = +

• (–3, –1) → x = –2 → (–)(+) = –

• (–1, ∞) → x = 0 → (+)(+) = +

We want ≥ 0 → positive or 0
✅ Answer: x ≤ –3 or x ≥ –1

🔷 3. Compound Inequalities

Sometimes two inequalities are combined:

🧩 Example 5:
Solve: –2 < 3x – 1 ≤ 5

Break into two:

1. –2 < 3x – 1 → 3x > –1 → x > –1/3

2. 3x – 1 ≤ 5 → 3x ≤ 6 → x ≤ 2
   ✅ Final Answer: –1/3 < x ≤ 2

🔷 4. Reasoning-style Symbol Inequalities

Used in reasoning sections, these involve coded relationships:

If A > B, B ≤ C, and C = D, then what is the relation between A and D?

👉 Translate step-by-step:
A > B
B ≤ C → So A > C
C = D → So A > D
✅ Answer: A > D

🔷 Important Rules and Tips

🧠 Rule 1: Inequality sign flips when you multiply or divide by a negative number.

🧠 Rule 2: Always write inequalities in standard form: variable terms on one side, constants on the other.

🧠 Rule 3: Quadratic inequality intervals change signs at roots.

🧠 Rule 4: For “> 0” or “< 0”, we exclude roots.
For “≥ 0” or “≤ 0”, we include roots.

🔶 Common MCQs

🧠 Q1: Solve: x² – 3x – 4 < 0
A. x < –1 or x > 4
B. –1 < x < 4
C. x ≤ –1 or x ≥ 4
D. x > –1

✅ Answer: B
Explanation: Roots = –1 and 4 → test signs → solution = (–1, 4)

🧠 Q2: Solve: 4x – 7 > 9
A. x < 4
B. x > 4
C. x > 2
D. x < 2

✅ Answer: B
Explanation: 4x > 16 → x > 4

🧠 Q3: Find the solution set: x² + 5x + 6 ≥ 0
A. x ≤ –2 or x ≥ –3
B. x ≤ –3 or x ≥ –2
C. x ≥ –3
D. x ≤ –3 or x ≤ –2

✅ Answer: B
Roots: –2 and –3 → intervals: (–∞, –3], [–2, ∞)